Group-stack

In algebraic geometry, a group-stack is an algebraic stack whose categories of points have group structures or even groupoid structures in a compatible way.^[1] It generalizes a group scheme, which is a scheme whose sets of points have group structures in a compatible way.

• A group scheme is a group-stack. More generally, a group algebraic-space, an algebraic-space analog of a group scheme, is a group-stack.
• Over a field k, a vector bundle stack ${\displaystyle {\mathcal {V}}}$ on a Deligne–Mumford stack X is a group-stack such that there is a vector bundle V over k on X and a presentation ${\displaystyle V\to {\mathcal {V}}}$. It has an action by the affine line ${\displaystyle \mathbb {A} ^{1}}$ corresponding to scalar multiplication.
• A Picard stack is an example of a group-stack (or groupoid-stack).

The definition of a group action of a group-stack is a bit tricky. First, given an algebraic stack X and a group scheme G on a base scheme S, a right action of G on X consists of

1. a morphism ${\displaystyle \sigma :X\times G\to X}$,
2. (associativity) a natural isomorphism ${\displaystyle \sigma \circ (m\times 1_{X}){\overset {\sim }{\to }}\sigma \circ (1_{X}\times \sigma )}$, where m is the multiplication on G,
3. (identity) a natural isomorphism ${\displaystyle 1_{X}{\overset {\sim }{\to }}\sigma \circ (1_{X}\times e)}$, where ${\displaystyle e:S\to G}$ is the identity section of G,

that satisfy the typical compatibility conditions.

If, more generally, G is a group-stack, one then extends the above using local presentations.

• Behrend, K.; Fantechi, B. (1997-03-01). "The intrinsic normal cone". Inventiones Mathematicae. 128 (1): 45–88. doi:10.1007/s002220050136. ISSN 0020-9910.